Simulated annealing is a probabilistic algorithm for approximately solving large combinatorial optimization problems. The algorithm can mathematically be described as the generation of a series of Markov chains, in which each Markov chain can be viewed as the outcome of a random experiment with unknown parameters (the probability of sampling a cost function value). Assuming a probability distribution on the values of the unknown parameters (the prior distribution) and given the sequence of configurations resulting from the generation of a Markov chain, we use Bayes's theorem to derive the posterior distribution on the values of the parameters. Numerical experiments are described which show that the posterior distribution can be used to predict accurately the behavior of the algorithm corresponding to the next Markov chain. This information is also used to derive optimal rules for choosing some of the parameters governing the convergence of the algorithm.

A special pattern of consecutive points on a control chart often suggests that the process is out of control due to a special cause. Usually, the probability of such a pattern is not difficult to calculate, but its relation to the rate of a false signal in a control chart is unclear. In this paper we give such a relation for a wide class of patterns that includes six of the eight patterns of special cause discussed in the AT&T Statistical Quality Control Handbook (1958). We derive a surprisingly simple formula for the rate of a false signal in terms of the pattern probability and use this result to improve on the test proposed in the handbook. We also use this formula to give a simple proof of a result of Wolfowitz (1944, Annals of Mathematical Statistics 15: 163–172) and Olmstead (1945, Annals of Mathematical Statistics 17: 24–33) on the monotone sequence. Finally, we give an approximation of the rate of a false signal for another wide class of patterns that covers the two remaining patterns of special cause in the AT&T handbook.

We define two new classes of multistate coherent systems by requiring, among other conditions, that their structure functions be Schur-concave (Schurconvex). The M + 1 performance levels of both the systems and their components are represented by the set [0, 1,…, M]. We present basic structural properties of the new classes. In particular, we study in some detail the number and form of the critical upper (lower) vectors to the various performance levels. We also present some probabilistic aspects of the new classes.

The main aim of this paper is to present two new results concerning the characterization of likelihood ratio and reversed hazard rate orders between largest order statistics from two sets of independent heterogeneous and homogeneous exponentiated generalized gamma distributed random variables. These characterization results complete and strengthen some previous ones in the literature.

In this paper, we study the pricing of vulnerable Asian options with liquidity risk. We employ general Lévy processes to capture the changes in the liquidity discount factors and the information processes of all assets. In the proposed pricing model, we obtain the closed-form pricing formula of vulnerable Asian options using the Fourier transform methods. Finally, the derived pricing formula is used to illustrate the effects of asymmetric jump risk, and the effects are relatively stable on (vulnerable) Asian options with different moneynesses.

In this article, we develop methods for estimating the expected time to the first loss in an Erlang loss system. We are primarily interested in estimating this quantity under light traffic conditions. We propose and compare three simulation techniques as well as two Markov chain approximations. We show that the Markov chain approximations proposed by us are asymptotically exact when the load offered to the system goes to zero. The article also serves to highlight the fact that efficient estimation of transient quantities of stochastic systems often requires the use of techniques that combine analytical results with simulation.

This paper considers stochastic comparison of parallel systems in terms of likelihood ratio order under scale models. We introduce a new order, the so-called q-larger order, and show that under certain conditions, the q-larger order between the scale vectors can imply the likelihood ratio order of parallel systems. Applications are given to the generalized gamma scale family.

We consider the dynamic pricing problem of perishable products in a system with a constant production rate. Potential demands arrive according to a compound Poisson process, and are price-sensitive. We carry out the sample path analysis of the inventory process and by using level-crossing method, we derive its stationary distribution given a pricing function. Based on the distribution, we express the average profit function. By a stochastic comparison approach, we characterize the pricing strategy given different customers willingness-to-pay functions. Finally, we provide an approximation algorithm to calculate the optimal pricing function.

We propose the use of the cluster distribution, derived from a negative binomial probability model, to estimate the probability of high-order events in terms of number of lines outaged within a short time, useful in long-term planning and also in short-term operational defense to such events. We use this model to fit statistical data gathered for a 30-year period for North America. The model is compared to the commonly used Poisson model and the power-law model. Results indicate that the Poisson model underestimates the probability of higher-order events, whereas the power-law model overestimates it. We use the strict chi-square fitness test to compare the fitness of these three models and find that the cluster model is superior to the other two models for the data used in the study.

This paper deals with evaluation and optimization of polling systems with time limits. Performance measures are evaluated with the power-series algorithm, a flexible technique for computing performance measures for multiqueue systems. The constant time limits are approximated by Erlang distributed variables. The algorithm is extended to compute derivatives of performance measures. This allows for optimization of cost functions with respect to the mean values of the time limits by gradient methods. Several properties of the optimal time limits are revealed by the numerical solution of various optimization problems.

Using the Laplace transform, we characterize, by means of necessary and sufficient conditions, the property that two life distributions are ordered in the sense of stochastic ordering, hazard rate ordering, backward hazard rate ordering, and likelihood ratio ordering.

Past research on polling systems has been quite restricted in the form of the paths followed by the server. This paper formulates a general, continuous model of such paths that includes closed walks on graphs. Customers arrive by a Poisson process and have general service times. The distribution of arrivals over the path is governed by an absolutely continuous, but otherwise arbitrary, distribution. The main results include a characterization of the stationary state distribution and explicit formulas for expected waiting times. The formulas reveal an interesting decomposition of the system into two components: a fluid limit and an M/G/1 queue.

In this paper, we extend the strong laws of large numbers and entropy ergodic theorem for partial sums for tree-indexed nonhomogeneous Markov chains fields to delayed versions of nonhomogeneous Markov chains fields indexed by a homogeneous tree. At first we study a generalized strong limit theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Then we prove the generalized strong laws of large numbers and the generalized asymptotic equipartition property for delayed sums of finite nonhomogeneous Markov chains indexed by a homogeneous tree. As corollaries, we can get the similar results of some current literatures. In this paper, the problem settings may not allow to use Doob's martingale convergence theorem, and we overcome this difficulty by using Borel–Cantelli Lemma so that our proof technique also has some new elements compared with the reference Yang and Ye (2007).

Probabilistic analyses of hashing algorithms usually assume that hash values are uniformly distributed over addresses. We study how one of the simplest schemes, hashing with linear probing, behaves in the nonuniform case. A simple measure μ of nonuniformity is the probability two keys hash to the same address, divided by this probability in the uniform case. It turns out that the effect of nonuniformity is to multiply mean search lengths by μ. For high loads, the longest search is multiplied by approximately μ also. Our theoretical results are asymptotics: simulations show good fits with predictions for mean search lengths, but bad fits for longest search lengths.

Let $\{Y_{1},\ldots ,Y_{n}\}$ be a collection of interdependent nonnegative random variables, with $Y_{i}$ having an exponentiated location-scale model with location parameter $\mu _i$, scale parameter $\delta _i$ and shape (skewness) parameter $\beta _i$, for $i\in \mathbb {I}_{n}=\{1,\ldots ,n\}$. Furthermore, let $\{L_1^{*},\ldots ,L_n^{*}\}$ be a set of independent Bernoulli random variables, independently of $Y_{i}$'s, with $E(L_{i}^{*})=p_{i}^{*}$, for $i\in \mathbb {I}_{n}.$ Under this setup, the portfolio of risks is the collection $\{T_{1}^{*}=L_{1}^{*}Y_{1},\ldots ,T_{n}^{*}=L_{n}^{*}Y_{n}\}$, wherein $T_{i}^{*}=L_{i}^{*}Y_{i}$ represents the $i$th claim amount. This article then presents several sufficient conditions, under which the smallest claim amounts are compared in terms of the usual stochastic and hazard rate orders. The comparison results are obtained when the dependence structure among the claim severities are modeled by (i) an Archimedean survival copula and (ii) a general survival copula. Several examples are also presented to illustrate the established results.

We describe and develop a variation on a layered multishift coupler due to Wilson that uses a slice sampling procedure to allow one to obtain potentially common draws from two different distributions. Our main application is coupling sample paths of Markov chains for use in perfect sampling algorithms. The coupler is based on slicing density functions and we describe a “folding” mechanism as an attractive alternative to the accept/reject step commonly used in slice sampling algorithms. Applications of the coupler are given to storage models and to auto-gamma distribution sampling.

A consecutive-k−out−of−n system is a graph with n vertices where the system fails if and only if some path of k consecutive vertices all fail. The assumption is made that qi, is the failing probability of the ith vertex and the states of the vertices are statistically independent. The problem is to find the best system, i.e., a set {q1, …,qn}, and its assignment to the vertices of the system graph that maximizes the reliability of the system, under the constraint that the product of qi's is a constant. We solve the problem for all lines and some cycles. We then extend our results to trees and show that for given n and k the best tree is a star and the worst is a line.

In this article, we develop a simple method to approximate the transient behavior of queueing systems. In particular, it is shown how singularity analysis of a known generating function of a transient sequence of some performance measure leads to an approximation of this sequence. To illustrate our approach, several specific transient sequences are investigated in detail. By means of some numerical examples, we validate our approximations and demonstrate the usefulness of the technique.

This paper studies the friendship paradox for weighted and directed networks, from a probabilistic perspective. We consolidate and extend recent results of Cao and Ross and Kramer, Cutler and Radcliffe, to weighted networks. Friendship paradox results for directed networks are given; connections to detailed balance are considered.